Turns out one can compute a bit more for hypersurfaces
$$\varphi \colon H_2(X, \mathbf{Z}) \times H^2_{\mathrm{dR}}(X/k) \to \mathbf{C} \qquad (\gamma, \omega) \longmapsto \int_\gamma \omega$$
Note, if $\gamma \in \operatorname{Pic} X^{al}$, then $\frac{1}{2\pi i}\int_\gamma \omega \in k^{al}$ for $\omega \in F^1 H^2_{\mathrm{dR}}(X/k)$.
If $\gamma = [C] \in H_2(X, \mathbf{Z})$ for a curve $C \subset X$ then from $\frac{1}{2\pi i}\left(\int_\gamma \omega\right)_{\omega \in F^1}$ one can construct an ideal $I_\gamma$ such that $I(C) \subsetneq I_\gamma$.